Bootstrap Unit Root Tests: Comparison and Extensions

Transcription

1 This is the accepted versio of the followig article: Palm, F.C., S. Smeekes ad J.-P. Urbai (2008). Bootstrap Uit-Root Tests: Compariso ad Extesios. Joural of Time Series Aalysis 29 (2), , which has bee published i fial form at Bootstrap Uit Root Tests: Compariso ad Extesios Fraz C. Palm, Stepha Smeekes, Jea-Pierre Urbai Departmet of Quatitative Ecoomics Uiversiteit Maastricht September 27, 2007 Abstract I this paper we study ad compare the properties of several bootstrap uit root tests recetly proposed i the literature. The tests are Dickey-Fuller or Augmeted DF-tests, either based o residuals from a autoregressio ad the use of the block bootstrap or o first differeced data ad the use of the statioary bootstrap or sieve bootstrap. We exted the aalysis by iterchagig the data trasformatios (differeces versus residuals), the types of bootstrap ad the presece or absece of a correctio for autocorrelatio i the tests. We show that two sieve bootstrap tests based o residuals remai asymptotically valid. I cotrast to the literature which focuses o a compariso of the bootstrap tests with a asymptotic test, we compare the bootstrap tests amog them usig respose surfaces for their size ad power i a simulatio study. This study leads to the followig coclusios: (i) augmeted DF-tests are always preferred to stadard DF-tests; (ii) the sieve bootstrap performs better tha the block bootstrap; (iii) differece-based tests appear to have slightly better size properties but residual-based tests appear more powerful. Keywords: bootstrap uit root tests, mote carlo, respose surface. JEL Codes: C15, C22 Correspodig author: Departmet of Quatitative Ecoomics, Uiversiteit Maastricht, P.O. Box 616, 6200 MD Maastricht, The Netherlads. f.palm@ke.uimaas.l. We thak Jeroe va de Berg for his assistace with our simulatios ad Robert Taylor, a aoymous referee ad participats at the Ecoometric Society World Cogress, Lodo, August 2005, for helpful commets ad suggestios. The usual disclaimer applies. 1

2 1 Itroductio Due to the good performace of the bootstrap i fiite samples for statioary processes, its applicatio to ostatioary series has recetly become icreasigly popular. I this paper we study ad compare the properties of some bootstrap uit root tests that have recetly bee proposed i the literature. We also itroduce some ew tests, show their first order asymptotic validity ad compare them to existig tests. The tests cosidered are Dickey-Fuller (DF) or Augmeted Dickey-Fuller (ADF) tests, either based o residuals from a autoregressio ad the use of the block bootstrap (Paparoditis ad Politis, 2003) or o first differeced data ad the use of the statioary bootstrap (Swese, 2003a) or sieve bootstrap (Psaradakis, 2001; Chag ad Park, 2003). As metioed, these papers differ i the way the bootstrap uit root tests. Besides showig the asymptotic validity, 1 all these papers compare the fiite sample performace of their test(s) to the asymptotic couterpart(s), ad the results are overall ecouragig. It is however less clear how these tests perform compared to each other. The goal of this paper is to fid out which tests perform best uder circumstaces to be give, ad which aspects of the tests determie their fiite sample performace. We will aalyse ad compare the asymptotic properties of these tests, ad we will also cosider Mote Carlo simulatios. We distiguish three mai features of the tests. The first feature is the actual test statistic. Some tests use the DF test, others the ADF. As the ADF statistic is asymptotically pivotal, whereas the DF is ot, we might expect a bootstrap ADF test to offer asymptotic refiemets over the bootstrap DF test ad asymptotic tests (Horowitz, 2001). 2 The secod feature is which series exactly should be resampled. Bootstrappig a ostatioary series directly is ot valid (Basawa et al., 1991). Therefore a statioary series has to be costructed first. Some tests use residuals from a first-order autoregressio of the series, others use first-differeces of the series. Swese (2003b) shows that power fuctios are the same for both cases if the iovatios are iid. However as show by Paparoditis ad Politis (2003, 2005), the use of differeces leads to poor behaviour of the bootstrap tests uder the alterative. The third feature is the time series bootstrap method that is employed. Some tests that we cosider use some form of the block bootstrap, i which blocks of (restricted) residuals are resampled. Other tests use the sieve bootstrap, that fits a AR model to the (restricted) residuals ad resamples the residuals of this AR model. The sieve bootstrap is somewhat easier to use ad performs better whe valid, but the block bootstrap is valid for more geeral processes. Curretly, to our kowledge o tests that use the sieve bootstrap based o residuals have bee show to be asymptotically valid for the Data Geeratig Processes (DGPs) cosidered i this paper. We adapt the sieve bootstrap tests by Psaradakis (2001) ad Chag ad Park 1 We call a test asymptotically valid if the bootstrap distributio uder the ull coverges to the asymptotic ull distributio. 2 Park (2003) shows that bootstrap ADF tests offer asymptotic refiemets uder the assumptio the errors are a fiite AR process with kow order. 2

3 (2003) by costructig them usig residuals istead of differeces ad show that these ew tests are asymptotically valid. As residual-based tests may have better properties uder the alterative tha differece-based tests, this is a importat extesio. With these results, all the tests cosidered i this paper have bee show to be asymptotically valid. A word o otatio. We deote weak covergece by d, covergece i probability by p ad almost sure covergece by a.s.. W(r) idicates a stadard Browia motio. As usual, we use the superscript to deote bootstrap quatities, both for bootstrap samples ad statistics calculated for bootstrap samples. Similarly, d idicates weak covergece of a bootstrap statistic coditioal o the origial series. The structure of the paper is as follows. I Sectio 2, we discuss the bootstrap uit root tests, highlight several features of these tests ad prove the asymptotic validity of the ew tests proposed. Sectio 3 cotais a extesive Mote Carlo simulatio aalysis of the various bootstrap uit root tests. The results are summarised usig respose surfaces. Sectio 4 cocludes. 2 The tests I this sectio we discuss several bootstrap uit root tests from a theoretical poit of view. 2.1 DF sieve bootstrap test Differece-based DF sieve bootstrap test: Psaradakis (2001) Psaradakis (2001) cosiders the followig DGP for the time series y t,t = 1,...,: y t = d t + v t, v t = ρv t 1 + u t, (1) where d t cosists of determiistic compoets. Three cases for the determiistic compoets are cosidered: the first case is without determiistics, d t = 0, the secod case is with oly a costat term, d µ t = δ 0, ad the third case is with costat term ad liear time tred, d τ t = δ 0 + δ 1 t. The process u t is assumed to satisfy the followig coditio with r = 4 ad s = 1: Assumptio 1 (i) the process u t is geerated by u t = j=0 ψ jε t j, with ε t a sequece of iid radom variables with E[ε t ] = 0, E[ε 2 t ] = σ2 ε > 0 ad E[εr t ] <. (ii) (A) Let ψ 0 = 1, js ψ j < ad j=0 ψ j 0. (B) j=0 ψ jz j is bouded, ad bouded away from zero for {z C : z 1}. 3

4 Note that this assumptio implies that u t is a ivertible liear process; see Phillips ad Solo (1992) for more details. We ca rewrite the model (1) ito the followig form y t = ρy t 1 + d t + u t, (2) where d t = γ 0 +γ 1 t := (1 ρ)δ 0 +ρδ 1 +(1 ρ)δ 1 t (i the first case δ 0 = δ 1 = 0, i the secod case δ 1 = 0). Psaradakis cosiders the DF coefficiet test (ˆρ 1) ad t-test i equatio (2) for testig ρ = 1. As stated above, the assumptios o the iovatios allow for a sieve bootstrap. Psaradakis (2001) furthermore eeds the followig assumptio o the order of the autoregressio: Assumptio 2 The order p of the autoregressive approximatio is such that p = p() as with p() = o((/l ) 1/4 ). The exact bootstrap procedure ca be described as follows. Bootstrap Test 1 (Psaradakis, 2001) 1. Fit a AR(p) model to û t, where û t = y t if the determiistic part cosists of at most a costat term, ad û t = y t 1 y t if the determiistic part cotais both a costat term ad a liear time tred, to obtai estimates ˆφ j, ad ˆε t, = û t ˆφ j, û t j, t = 1 + p,...,. 2. Geerate a iid sample ε t, by drawig radomly with replacemet from ˆε t, ( p) 1 +p ˆε t,. 3. Costruct bootstrap errors by the recursio u t, = ˆφ j, u t j, + ε t,. (3) 4. The bootstrap sample y t, is geerated recursively by y t, = y t 1, + u t, i case of o determiistic compoets or a itercept oly, ad by yt, = 1 y t + yt 1, + u t, i case of a costat term ad a liear tred. 4

5 5. Calculate the DF coefficiet test ad t-test usig the bootstrap sample for the previously specified determiistic specificatio. 6. Repeat steps 2 to 5 B times to fid the bootstrap distributios where B deotes the umber of bootstrap replicatios. Psaradakis (2001) suggests to estimate the AR(p) model i step 1 usig the Yule-Walker equatios to esure that the geerated iovatios u t, admit a oe-sided MA( ) represetatio. The asymptotic distributio of the bootstrap statistics uder the ull is show to be the same as the asymptotic distributio of the origial DF statistics. Note that although the limitig distributios cotai uisace parameters, this does ot matter for the bootstrap approach as the critical values for testig are based o the (empirical) distributios of the bootstrap tests that ca be approximated by simulatio with ay accuracy desired Residual-based DF sieve bootstrap test: Psaradakis modified The test we propose here is very similar to the test by Psaradakis (2001), except that it is based o residuals. Paparoditis ad Politis (2005) have proposed a ADF coefficiet test, ad we costruct our test i the same way as they do. We will show that our test is asymptotically valid whe cosiderig the assumptios made by Psaradakis (2001). The ew algorithm differs from that for the tests by Psaradakis (2001) oly i step 1: Bootstrap Test 2 (Residual-based DF sieve bootstrap procedure) Replace step 1 from Bootstrap Test 1 by calculatig the residuals from the regressio ˆε t, = ỹ t ˆρ ỹ t 1 ˆφ j, ỹ t j, t = 1 + p,...,, (4) where ỹ t = y t i the case of a (possibly zero) itercept ad ỹ t = y t ˆγ 0 ˆγ 1 t i the case of a liear tred, ad ˆγ 0 ad ˆγ 1 are the correspodig OLS estimates. The ext theorem shows that, uder the assumptios give above, the bootstrap distributios coverge to the same limit distributio as the stadard test statistics: Theorem 1 Let τ = (ˆρ 1) ad t be the coefficiet ad t-statistic, respectively, that follow from Bootstrap Procedure 2. Let σu 2 = E[u2 t ] ad σ2 = lim 1 E[( u t) 2 ]. Uder Assumptios 1 with r = 4 ad s = 1 ad 2, we have that τ t 1 d 0 W(r)dW(r) + (σ2 σu)/2σ W(r)2 dr 1 d i probability 0 W(r)dW(r) + (σ2 σu)/2σ 2 2 ( (σu 2/σ2 ) ) 1 1/2 i probability. 0 W(r)2 dr 5

6 where W(r) is a stadard Browia motio o [0,1]. Proof : See Appedix A. We have show that the DF sieve test as costructed by Psaradakis (2001) remais asymptotically valid if it is based o residuals istead of differeces. 2.2 ADF sieve bootstrap test Differece-based ADF sieve bootstrap test: Chag ad Park (2003) Chag ad Park (2003) cosider the DGP y t = ρy t 1 + u t, (5) where u t = j=0 ψ jε t j. They employ Assumptio 1 with r 4 ad s 1. For the order of the autoregressive approximatio, Chag ad Park (2003) cosider two differet assumptios: Assumptio 3 Let p() ad p() = o( κ ) with κ < 1 2 as. The followig assumptio is stroger. Assumptio 4 Let p() = c κ for some costat c ad 1/rs < κ < 1 2. The bootstrap procedures of Chag ad Park (2003) ad Psaradakis (2001) are very similar: Bootstrap Test 3 (Chag ad Park, 2003) Follow the same steps as i Bootstrap Test 1, but oly for the determiistic specificatio d t = 0. Replace step 5 by 5. Calculate the ADF coefficiet statistic (1 p ˆφ j, ) 1 (ˆρ 1) ad the correspodig t-statistic 3 from the ADF regressio y t, = ρ y t 1, + φ j yt j, + ε t. Chag ad Park (2003) show that their bootstrap tests coverge to the same asymptotic distributios uder the ull as the asymptotic tests. The covergece is show to hold almost surely uder the strog assumptios, ad i probability uder the weaker assumptios. They claim that their tests are also valid whe applied to demeaed or detreded data, but they do ot provide ay further aalysis. 3 Chag ad Park suggest usig ˆσ 2 ε, (calculated from the origial sample) for the t-test istead of ˆσ 2 ε, (calculated from the bootstrap sample), although both are appropriate. Similarly, it is possible to use 1 p ˆφ j, for the coefficiet test. 6

7 2.2.2 Residual-based ADF sieve bootstrap test: Chag ad Park modified Similar to the previous sectio, we costruct a residual-based test that is based o the test by Chag ad Park (2003) ad resembles the residual-based ADF test of Paparoditis ad Politis (2005) strogly. Bootstrap Test 4 (Residual-based ADF sieve bootstrap test) Replace step 1 from Bootstrap Test 3 by calculatig the residuals from a ADF regressio as i the followig equatio ˆε t, = y t ρ y t 1 ˆφ j, y t j, t = 1 + p,...,. (6) The ext theorem shows that, uder the assumptios give above, the bootstrap distributios coverge to the same limit distributios as the asymptotic test statistics. Theorem 2 Let τ ad t be the bootstrap coefficiet statistic ad t-statistic, respectively, that follow from Bootstrap Test 4. Let Assumptios 1 with r 4 ad s 1 ad 3 hold. The τ 1 d 0 W(r)dW(r) 1 0 W(r)2 dr i probability, t Proof : See Appedix A. d 1 0 ( W(r)dW(r) ) 1 1/2 i probability. 0 W(r)2 dr We have show that the ADF sieve test as costructed by Chag ad Park (2003) is also asymptotically valid if it is based o residuals. I Theorem 2 we have obtaied covergece i probability whereas Chag ad Park (2003) proved a.s. covergece for their strog assumptios. By imposig the uit root restrictio differece-based tests rely o statioary series for which a.s. covergece holds. Although ot imposig the uit root whe applyig the sieve bootstrap is certaily a drawback, our result is worthwhile as it does provide justificatio for usig a residual-based sieve bootstrap, eve if it is ot the same justificatio as Chag ad Park (2003) provide for their tests. For fiite order AR(p) processes, Paparoditis ad Politis (2005) show that uder fixed alteratives the bootstrap distributio of the residual-based sieve bootstrap coefficiet test is the same as that uder the ull. For the differece-based sieve bootstrap the distributio uder the ull differs from that uder the alterative for the coefficiet test, but ot for the t-test. This results i a loss of power for the differece-based sieve bootstrap coefficiet test. For the t-tests, both methods are asymptotically equivalet. For AR( ) processes, Paparoditis ad Politis (2005) do ot discuss the residual-based sieve test, but they state 7

8 that the differece-based sieve bootstrap is iappropriate as the differeced process is ot ivertible if the alterative is true. 2.3 (A)DF block bootstrap test Residual-based (A)DF block bootstrap test: Paparoditis ad Politis (2003) Paparoditis ad Politis (2003) propose a block bootstrap method to test for uit roots. Their method, the residual-based block bootstrap (RBB), is a block bootstrap method applied to residuals of a regressio of the series y t o its first lag. We first state the assumptios uder which the RBB is appropriate. Two sets of assumptios are cosidered, such that oe of these should be satisfied by the process y t to validate the use of the RBB. Paparoditis ad Politis (2003) cosider the process y t = α + ρy t 1 + u t where if α 0 there is a drift uder the ull of ρ = 1. Paparoditis ad Politis (2003) cosider two sets for u t. The first is that Assumptio 1 (i) ad (ii)(a) hold with r = 4 ad s = 1 uder the ull. Uder the alterative these assumptios should hold for y t as well. Uder the additioal assumptio (ii)(b) the process is ivertible as well. This assumptio is similar to those Psaradakis ad Chag ad Park employ. The secod assumptio that Paparoditis ad Politis (2003) use, is that u t is strog mixig: Assumptio 5 For each value of ρ, the series u t is strog mixig ad satisfies the followig coditios: E[u t ] = 0, E u t r < for some r > 2, f u (0) > 0, where f u deotes the spectral desity of u t, i.e., f u (λ) = h= γ u(h)exp(iλh) ad γ u (h) = E[u t u t+h ]. Furthermore, k=0 α(k)1 2/r <, where α( ) deotes the strog mixig coefficiet of u t. I cotrast to the coditio eeded for the sieve bootstrap, oe should ote that the geeratig process of u t does ot have to be belog to the class of liear processes to satisfy this coditio. Hece, we see here a class of processes (possibly o-liear) for which the block bootstrap is valid but the sieve bootstrap is ot. The procedure proposed by Paparoditis ad Politis (2003) ca be described as follows. Bootstrap Test 5 (Paparoditis ad Politis, 2003) 1. Calculate the cetred residuals ũ t, = û t, 1 1 j=2 where ρ is a cosistet estimator of ρ. û t, = (y t ρ y t 1 ) 1 1 (y t ρ y t 1 ), 2. Choose the block legth b, ad draw poits i 0,i 1,...,i k 1, where k = ( 1)/b, 4 from the uiform distributio o the set {1,2,..., b}. These poits will serve as the 4 The bootstrap sample y t will have total legth l = kb + 1. j=2 8

9 begiig poits of the blocks of cetred residuals: y t, = { y 1 for t = 1 α + y t 1, + ũ i m+s, for t = 2,3,...,, (7) where m = (t 2)/b, s = t mb 1, ad α is a drift parameter that is either set equal to zero or it is a cosistet estimator of α. 3. From the bootstrap series y t, compute the desired statistics. 4. Repeat steps 2 to 3 B times to fid the bootstrap distributio. Although most bootstrap uit root tests are based o differeces, Paparoditis ad Politis (2003) formally show that the residual-based block bootstrap coefficiet test performs well asymptotically both uder the ull ad uder cotiguous alteratives whereas the asymptotic distributio of the differece-based block bootstrap (DBB) statistic differs from that of the RBB statistic uder the alterative, leadig to a loss of power of the DBB test. Moreover, the covergece rate for this DBB bootstrap test is slower uder the alterative tha for the RBB test. For fixed alteratives, the slower rate of covergece leads to a loss of power of the DBB test compared to the RBB test. For sequeces of 1 local alteratives, the two tests have the same power. I step 1 of the bootstrap procedure, ρ should be a cosistet estimator of ρ. Furthermore, it is required that ρ is o p (1) if ρ 1, O p ( 1 ) if ρ = 1 ad α = 0, ad O p ( 3/2 ) if ρ = 1 ad α 0. May estimators satisfy these coditios. Paparoditis ad Politis (2003) focus o what they call the least squares (LS) estimator, which is just the DF estimator, ad the ADF estimator (they call this the DF estimator). For the validity of the ADF estimator the additioal coditio (ii)(b) is eeded to esure ivertibility. They prove the cosistecy of the RBB for the DF coefficiet test ad the ADF coefficiet test. For models where α = 0, they recommed to use the OLS estimator of ρ i y t = α + ρy t 1 + u t (8) or the ADF equivalet 5 as ρ, which is used to costruct the residuals. I the secod step α is set to zero, as there should be o drift. They also recommed for the RBB ADF test to use the block bootstrap 6 of y t y t 1 directly as lagged differeces istead of y t, y t 1,. For both the tests without determiistic compoets ad the tests with a costat icluded, the cosistecy of the DF ad ADF RBB tests is proved. For the case of α 0, Paparoditis ad Politis (2003) recommed usig the same estimator for ρ as before but settig α = α where α is the estimator of α i (8). They prove the 5 Depedig o which uit root test is performed. 6 Usig the same blocks as for the residuals. 9

10 cosistecy of the DF coefficiet test with costat ad tred ad claim the cosistecy of the correspodig ADF test ca be established similarly Differece-based (A)DF block bootstrap test: Paparoditis ad Politis (2003) Agai we cosider a alterative versio of the tests by Paparoditis ad Politis (2003). As the origial tests are based o residuals, the modified tests will be based o differeces. The ew procedure simply replaces ρ by 1. Paparoditis ad Politis (2003) already showed the asymptotic validity of these alterative tests (also see the discussio of power above). 2.4 DF statioary bootstrap test Differece-based DF statioary bootstrap test: Swese (2003a) Swese (2003a) cosiders a uit root test without determiistic compoets based o the statioary bootstrap of Politis ad Romao (1994). He assumes the DGP y t = ρy t 1 + u t with the followig assumptios o u t. Assumptio 6 (i) The process u t is strictly statioary with E[u t ] = 0 for all t. (ii) If γ(k) = E[u t u t+k ], the γ 0 + r=0 rγ(r) < (iii) r,s,t κ 4(r,s,t) = K < where κ 4 (r,s,t) is the fourth cumulat of the distributio of (u j,u j+r,u j+r+s,u j+r+s+t ). Assumptio (iii) is used to esure that the variace of 1 t u2 t teds to zero ad implies that σ 2 ca be cosistetly estimated. Uder these coditios Swese (2003a) proves the cosistecy of the DF tests without determiistic compoets based o the statioary bootstrap. The coditios eeded are sigificatly weaker tha those eeded for the sieve bootstrap. The algorithm ca be described as follows: Bootstrap Test 6 (Swese, 2003) 1. Compute cetred differeces ũ t = y t ( 1) 1 y j. 2. Apply the statioary bootstrap of Politis ad Romao (1994) to the cetred residuals to obtai bootstrap errors u t,: j=2 10

11 (a) Draw the idex of the startig poits of the blocks, i 1,i 2,..., from the uiform distributio P(t 1 = t) = 1,t = 1,...,. Let p L be a fixed umber betwee 0 ad 1. Draw the legth of the blocks b 1,b 2,... from the geometric distributio P(b 1 = l) = (1 p L ) l 1 p L. The expected block legth is 1/p L. (b) Form blocks usig the draw startig poits ad block legths. For block m + 1 we have u t, = ũ i m+1 +t b(m) 1 (9) where t = b(m) + 1,...,b(m) + b m+1 ad b(m) = m b j. (c) Stop after geeratig B blocks if l B = B b j. Lay the blocks ed-to-ed i the order sampled, ad cut off the resultig series u 1,,...,u l B, at u, if l B >. 3. Costruct the bootstrap sample y t, with the recursio y t, = y t 1, + u t,. 4. Compute the bootstrap DF coefficiet ad t-statistic. 5. Repeat steps 2 to 4 B times to fid the bootstrap distributio Residual-based DF statioary bootstrap test: Parker, Paparoditis, ad Politis (2006) Agai, we also cosider a modified versio of these tests. Here we base the modified versio o residuals istead of differeces. Istead of the cetred differeces we calculate i step 1 cetred residuals as i the bootstrap procedure of Paparoditis ad Politis (2003). This test has bee proposed by Parker et al. (2006) who also show its asymptotic validity. 2.5 Summary of tests cosidered Whe comparig these tests, we will maily focus o three aspects: whether differeces or residuals are used, the bootstrap method ad the test statistic. Table 1 summarises all the test statistics ad their mai features. A ote o the otatio: we use τ for a coefficiet test ad t for a t-test. The first subscript idicates the bootstrap method: S stads for sieve bootstrap, B for block bootstrap, ad St for statioary bootstrap; the secod subscript idicates whether a test is based o differeces (d) or residuals (r). A superscript a states that the test is a augmeted DF test. [Table 1 about here.] 3 Fiite sample performace: Mote Carlo results We aalyse ad compare the fiite sample behaviour of the tests by Mote Carlo simulatios. 11

12 3.1 Mote Carlo setup We geerate a series y t,t = 1,...,, accordig to the recursio y t = ρy t 1 + u t, y 0 = 0, (10) where differet values for ρ are used: 1, 0.99, 0.95, 0.9 ad 0.8. We let u t be geerated by a ARMA(1,1) process: u t = φu t 1 + ε t + θε t 1, (11) where ε t IN(0,1). The values used for φ ad θ vary from -0.8 to As sample sizes we cosider = 50, 100 ad 250. We use three differet sigificace levels: 0.01, 0.05 ad All experimets will be based o 5000 simulatios ad 999 bootstrap replicatios. All simulatios are performed usig GAUSS 6.0. AIC is used to select the lag legth for the sieve bootstrap. We estimate the AR(p) models by OLS. 8 For the lag legth i the ADF tests we use the modified AIC by Ng ad Perro (2001), both outside ad iside the bootstrap procedures. For the block legth we choose fixed umbers: 5 for = 50, 8 for = 100 ad 15 for = 250. The fact that there is o easy way to estimate block legths remais a problem. We perform two sets of simulatios with these models. The first set cosiders the tests based o models without determiistic compoets. I the secod set of simulatios the DGPs remai uchaged but the tests are based o models with a costat ad a tred. These extesios are ot discussed i all papers, so that ot all tests we cosider have bee show to be theoretically valid. For the ADF test of Paparoditis ad Politis (2003), we follow their istructios o how to hadle the test allowig for a tred. Chag ad Park (2003) idicate that their tests ca be applied for the model with tred by applyig the bootstrap test to the detreded data. We detred both the origial series (by OLS) ad the bootstrap series. 9 For the test proposed by Swese (2003a), determiistic compoets are added i the same way as i Psaradakis (2001). 10 Note however that this is i fact ot ecessary, as the tests applied are actually ivariat to the determiistic compoets preset i the DGP, provided sufficiet determiistics are icluded i the test regressio. Therefore, the bootstrap test statistics are also ivariat to the determiistics i the bootstrap DGP as log they are correctly specified i the bootstrap test regressio. 7 Specific values used for (φ, θ) are: (0,0), ( 0.8, 0), ( 0.4,0), (0.4, 0), (0.8, 0), (0, 0.8), (0, 0.4), (0, 0.4), (0,0.8), (0.4, 0.4), ( 0.4, 0.4). 8 The estimated AR(p) model may ot be ivertible. A solutio could be to impose a root boud as i Burridge ad Taylor (2004). This is however maily importat for empirical work, as i a large simulatio study as ours the umber of cases i which the estimated process is ot ivertible, is very small. 9 It is crucial to detred the bootstrap series as well, otherwise the bootstrap distributio will ot coverge to the correct asymptotic distributio. 10 More efficiet detredig methods are ot cosidered i this paper. 12

13 The large umber of DGP s ad tests statistics i our simulatios leads to a huge umber of results that is rather hard to aalyse i stadard tables. We circumvet this problem by estimatig differet respose surfaces for the rejectio frequecies observed i our simulatios for each of the test statistics. Because the empirical rejectio frequecy ˆP lies betwee 0 ad 1, we use the followig trasformatio: ( ) ˆP L( ˆP) = l 1 ˆP. (12) The depedet variable is L( ˆP). As explaatory variables we cosider several fuctios of the omial level ad the parameters i the uderlyig DGP. We will provide more details below. The specific form of the respose surfaces is test specific. To avoid legthy specificatio searches, we rely o PcGets (Hedry ad Krolzig, 2001) to select the most appropriate specificatio from a large set of possible variables. The reported stadard errors are White s heteroscedasticity cosistet stadard errors. Apart from the coefficiet estimates ad their stadard errors, the adjusted R 2 of the regressio is also reported. 3.2 Results I this sectio we will give the mai fidigs of our simulatio study. We focus here o the results for the tests without determiistic treds. The results for the tests allowig for determiistic treds will be briefly discussed below. 11 Size Tables 2 give a summary of the respose surfaces for the size. We cosider the followig respose surface for the size: L( ˆP i ) = β 1 L(P a,i ) + β 2 f(l(p a,i),φ i,θ i, i ) + ν i, i = 1,...,M, (13) where P a is the omial size of the test, f(l(p a,i ),φ i,θ i, i ) is a vector of fuctios (all of order O( 1/2 ) ad O( 1/2 )) of L(P a ), the ARMA parameters φ i ad θ i ad the sample size i ad ν i deotes a disturbace. The umber of simulatio experimets M is 99. The term β 2 f( ) captures the deviatios of the actual size from the omial size as a fuctio of the parameters of the DGP ad sample size. β 1 L(P a,i ) gives a idicatio of the asymptotic size of the tests. Whe β 1 is equal to 1, the empirical size of the test is equal to the omial size for large. The table gives the estimate of β 1 ad its stadard error, as well a measure of the fit. [Table 2 about here.] 11 The collectio of all simulatio results, respose surfaces ad graphical aalyses is available o the Iteret: 13

14 Several thigs ca be see from the tables. We see that for some tests β 1 is sigificatly differet from 1, although for most it is close to it. The estimates for the residual-based sieve tests are all ot sigificatly differet from oe. Most of the other estimates are differet from oe, where especially the estimates for the DF test are far away from oe. Note that these are all DF tests, as opposed to the ADF tests for which β 1 is much closer to 1. The coefficiet ad t-tests appear to have similar size i most cases. For the block tests, the value for β 1 is higher for the differece-based versio tha the residual-based versio, which idicates that i geeral the residual-based block tests give higher rejectio frequecies tha differece-based block tests. We also see for all tests that the fit icreases whe we iclude variables of higher order. Especially the icrease i the fit from the first settig to the secod is oticeable. This shows that all tests suffer from fiite-sample distortios, although some more tha others. As ca be clearly see, the adjusted R 2 for the regressio o oly the omial size is much higher for the sieve tests tha for the block tests. This shows that the (especially ADF) sieve tests suffer less from fiite sample distortios tha the other tests. [Figure 1 about here.] [Figure 2 about here.] Figure 1 ad 2 show graphs of the fitted size plotted agaist the autoregressive ad movigaverage parameters φ ad θ. As omial level we take 0.05 ad as sample size we take 100. The gree area idicates a size betwee 0.03 ad 0.07, the blue area idicates a size below that rage ad the red area above that rage. The fitted trasformed sizes are calculated from the respose surfaces (13) for specific values φ 0, θ 0, 0 ad P a,0, substitutig estimates ˆβ 1 ad ˆβ 2 for β 1 ad β 2 respectively ad droppig the disturbace ν i. Next we apply the iverse of the L( ) trasformatio to the fitted values to obtai the fitted size. As well as cofirmig what the tables tell us, the graphs show how the AR ad MA parameter ifluece the empirical size. For all the tests, we see the well-kow size distortios for a large egative MA parameter. The extet of this size distortio differs however. The statioary bootstrap tests ad the DF block tests have massive size distortios, that also icrease whe the AR parameter becomes large ad egative. We see that these tests are much more sesitive to the values of φ ad θ, as they also exhibit a large udersize for large positive values. The ADF block tests maily exhibit large udersize, especially for large absolute values of φ ad θ. The sieve tests ca be see to perform quite well; especially the ADF sieve tests, for which the graphs are quite flat ad i the correct rage. We ca also see that i geeral residual-based tests have higher rejectio frequecies tha the differece-based tests, except for the ADF sieve tests where both perform equally well. 14

15 Power Tables 3 ad 4 give summaries of the respose surfaces for the power. We choose to report oly uadjusted power as we feel this is the most relevat, because this is what matters i practice. We ow estimate the followig respose surface: 3 L( ˆP i ) = β 0 +β 1 L(P a,i )+ β 2,k (ρ i 1) k +β 3f(ρ i,l(p a,i ),φ i,θ i, i )+ν i, i = 1,...,M. (14) k=1 The umber of simulatio experimets M is 396. Agai all variables i f(ρ,l(p a,i ),φ i,θ i, i ) are either of order O( 1/2 ) or O( 1 ). So i this case the asymptotic behaviour ca be deduced from β a = (β 0,β 1,β 2 ). The tables give the estimates plus stadard error for the O(1) variables ad a measure of the fit. Agai we see that the fit icreases whe we add higher order terms. [Table 3 about here.] [Table 4 about here.] I Figures 3 to 4 we give power curves for varyig sample sizes. These plots are derived from the respose surfaces i the same way as the surface graphs for the size. For all cases, we have take φ = θ = Most of the graphs show that the residual-based tests have higher power tha the differece-based tests. However, as we also foud that the residual-based tests have larger size distortios tha the differece-based tests i geeral, the higher power will partly be caused by the size distortios. I that respect, we see that the power differece betwee residual-based ad differece-based ADF sieve tests is quite small, while for these tests the behaviour uder the ull of residual-based tests ad differece-based tests was also comparable. Hece, if there is a power advatage for residual-based tests, it is oly small. [Figure 3 about here.] [Figure 4 about here.] Determiistic treds The tests allowig for determiistic treds give qualitatively similar results as the oes described above. All tests perform worse, however the effect of icludig the determiistic treds where i fact oe are eeded is similar for all tests. Power becomes lower i geeral, ad size seems to fluctuate more for differet AR ad MA parameters. 4 Coclusio We have aalysed the behaviour of a set of bootstrap uit root tests i fiite samples. Moreover, we have show the validity of two procedures that tur out to work well i fiite samples. 12 Ureported results show the depedece of the empirical power o φ ad θ is similar as i the case of the size. 15

16 From our simulatio study we ca draw several coclusios. First, ADF tests clearly perform better tha DF tests, which is what we expected from our discussio about asymptotically pivotal statistics. We do ot observe a clear differece betwee the coefficiet tests ad t-tests. Secod, it seems that sieve tests perform better i terms of size tha block tests for ARMA models, which is i lie with the results for statioary series. We also see that the statioary bootstrap test performs worse i terms of size tha the block bootstrap. Added to this, there is also a practical reaso to use the sieve bootstrap. The selectio of the lag legth ca be doe quite easily, ad appears to work if based o a iformatio criterio like AIC or modified AIC. O the other had, choosig the block legth o the basis of ituitio is difficult, ad there exist o satisfactory methods for it. Takig all this ito accout, for our set of models the sieve bootstrap is preferable over the block bootstrap. Third, the choice betwee differece-based tests ad residual-based tests is less obvious. While the residual-based tests have higher power tha the differece-based tests, these tests also have higher size distortios. However, whe we cosider ADF sieve bootstrap test, the residual-based tests perform similarly as the differece-based tests both i terms of size ad i terms of power. These fidigs are i lie with the simulatio results reported i the previous studies metioed i the itroductio, i the way the tests perform for differet ARMA parameters. Our fidigs however allowed us to systematically compare existig ad ewly proposed tests. O the basis of previous studies oly, it was ot clear how the various tests compared. Summarisig, for the type of processes cosidered, the ADF sieve tests perform best i our simulatio study. Therefore, for settigs comparable to ours, we ca recommed to use either the tests by Chag ad Park (2003) or the ADF sieve tests based o residuals that we proposed. For other types of processes, allowig for broke treds, heteroskedasticity, etc., further research is eeded. Refereces Basawa, I. V., A. K. Mallik, W. P. McCormick, J. H. Reeves, ad R. L. Taylor (1991). Bootstrappig ustable first-order autoregressive processes. Aals of Statistics 19, Bühlma, P. (1995). Movig-average represetatios of autoregressive approximatios. Stochastic Processes ad their Applicatios 60, Bühlma, P. (1997). Sieve bootstrap for time series. Beroulli 3, Burridge, P. ad A. M. R. Taylor (2004). Bootstrappig the HEGY seasoal uit root tests. Joural of Ecoometrics 123, Chag, Y. ad J. Y. Park (2002). O the asymptotics of ADF tests for uit roots. Ecoometric Reviews 21,

17 Chag, Y. ad J. Y. Park (2003). A sieve bootstrap for the test of a uit root. Joural of Time Series Aalysis 24, Hedry, D. F. ad H.-M. Krolzig (2001). Automatic Ecoometric Model Selectio. Lodo: Timberlake Cosultats Press. Horowitz, J. L. (2001). The bootstrap. I J. J. Heckma ad E. E. Leamer (Eds.), Hadbook of Ecoometrics, Volume 5, Chapter 52, pp Amsterdam: North Hollad Publishig. Ng, S. ad P. Perro (2001). Lag legth selectio ad the costructio of uit root tests with good size ad power. Ecoometrica 69, Paparoditis, E. ad D. N. Politis (2003). Residual-based block bootstrap for uit root testig. Ecoometrica 71, Paparoditis, E. ad D. N. Politis (2005). Bootstrappig uit root tests for autoregressive time series. Joural of the America Statistical Associatio 100, Park, J. Y. (2002). A ivariace priciple for sieve bootstrap i time series. Ecoometric Theory 18, Park, J. Y. (2003). Bootstrap uit root tests. Ecoometrica 71, Parker, C., E. Paparoditis, ad D. N. Politis (2006). Uit root testig via the statioary bootstrap. Joural of Ecoometrics 133, Phillips, P. C. B. ad V. Solo (1992). Asymptotics for liear processes. Aals of Statistics 20, Politis, D. N. ad J. P. Romao (1994). The statioary bootstrap. Joural of the America Statistical Associatio 89, Psaradakis, Z. (2001). Bootstrap tests for a autoregressive uit root i the presece of weakly depedet errors. Joural of Time Series Aalysis 22, Swese, A. R. (2003a). Bootstrappig uit root tests for itegrated processes. Joural of Time Series Aalysis 24, Swese, A. R. (2003b). A ote o the power of bootstrap uit root tests. Ecoometric Theory 19, A Proofs Our proofs are adaptatios of the proofs of Psaradakis (2001) ad Chag ad Park (2003) (which i tur depeds o Park (2002)). We oly elaborate where our proofs differ from theirs due to the use of residuals istead of differeces. To be specific, the residuals to be resampled i our tests are costructed as ˆε t, = y t ˆρ y t 1 ˆφ j, y t j, (15) 17

18 where ˆρ, ˆφ 1,,..., ˆφ p, are the OLS estimates from the (augmeted Dickey-Fuller) regressio of y t o y t 1, y t 1,..., y t p. The residuals to be resampled i the tests of Psaradakis (2001) ad Chag ad Park (2003) are costructed as ε t, = y t φ j, y t j, (16) where φ 1,,..., φ p, are the OLS (or Yule-Walker) estimates from the regressio of y t o y t 1,..., y t p. Let φ = ( φ 1,,..., φ p, ), ˆφ = (ˆφ 1,,..., ˆφ p, ) ad x p,t = ( y t 1,..., y t p ). The ˆφ ad φ are related by ( ) 1 ( ˆφ = φ ) + (ˆρ 1) x p,t x p,t x p,t y t 1 (17) as i Chag ad Park (2002, Proof of Lemma 3.5). From this we ca deduce that ˆφ j, = φ j, + O p ( 1 ) (18) uder the ull hypothesis of a uit root. Oe cosequece of ot imposig the uit root restrictio is that ρ has to be estimated so that we are oly able to show some results i terms of covergece i probability istead of almost sure covergece. Note that we oly focus o the bootstrap distributios uder the ull, i lie with most of the literature ad specifically the papers that we base the ew tests o. To aalyse power properties, oe eeds to look at the bootstrap distributio uder alteratives as well. I the mai text we discuss the fidigs of Paparoditis ad Politis (2005), who cosider the power of these type of tests. A.1 Proof of Theorem 1 I order to prove this theorem we eed the followig lemmas: Lemma 1 Suppose Assumptios 1 (with r = 4 ad s = 1) ad 2 hold. The E [(ε t, )2w ] = E[(ε t ) 2w ] + o p (1) for w = 1, 2. (19) Proof of Lemma 1 We adapt the proof of Bühlma (1997, Proof of Lemma 5.3). First ote that E [(ε t,) 2w ] = ( p) 1 (ˆε t, ˆε ( ) ) 2w, (20) where ˆε ( ) = ( p) 1 ˆε t,. We first show that ˆε ( ) = o p(1). (21) 18

19 Note that uder the ull ε t = y t φ j y t j ˆε t, = y t ˆρ y t 1 ˆφ j, y t j. (22a) (22b) The we write ˆε ( ) = ( p) 1 (ε t ε t + ˆε t, ) = ( p) 1 = ( p) 1 ε t ( y t [ φ j y t j ) + (y t ˆρ y t 1 ε t ( y t (y t ˆρ y t 1 )) (ˆφ j, φ j ) y t j + = ( p) 1 (A t, + B t, + C t, + D t, ) j=p+1 φ j y t j ] ˆφ j, y t j ) (23) Hece it has to be show that the four right had side compoets i (23) are o p (1). It is trivial that ( p) 1 A t, ad ( p) 1 D t, are o p (1). Next we tur to B t, : Uder the ull 1 ˆρ = O p ( 1 ) (Chag ad Park, 2002), so that B t, = y t + y t ˆρ y t 1 = (1 ˆρ )y t 1 (24) ( p) 1 (1 ˆρ )y t 1 = (1 ˆρ )( p) 1 y t 1 = o p (1). (25) Fially, we cosider C t,. By the Cauchy-Schwartz iequality, ( p) 1 (ˆφ j, φ j ) y t j 1/2 (ˆφ j, φ j ) 2 ( p) 1 ( y t j ) 2 1/2. (26) As ˆφ j, φ j = O p ( (l /) 1/2 ) +o ( p 1) (Chag ad Park, 2002, Lemma 3.5) ad p() = o ( (/ l) 1/4), 19

20 we have (ˆφ j, φ j ) 2 1/2 = {O p ((l /) 1/2) + o ( p 1)} 2 = {O p ((l /) 1/4) + o p (p 1 (l /) 1/2) + o ( p 2)} 1/2 = [O p (p(l/) 1/4) + o p ((l/) 1/2) + o ( p 1)] 1/2 ( = [o p (l/) 1/2) 1/2 + o(p )] 1. 1/2 (27) Therefore, ( p) 1 C t = Havig show (21), we ow eed to show that As i (23), write [o p ( (l/) 1/2) + o(p 1 )] 1/2 Op (p 1/2 ) = o p (1). (28) ( p) 1 (ˆε t, ) 2w = E[(ε t ) 2w ] + o p (1). (29) ˆε t, = ε t ( y t (y t ˆρ y t 1 )) (ˆφ j, φ j ) y t j + φ j y t j = A t, + B t, + C t, + D t,. j=p+1 (30) Usig the argumets i (24) to (28), we have ( p) 1 B t, 2w = o p (1), (31a) ( p) 1 C t, 2w = o p (1), (31b) ( p) 1 D t, 2w = o p (1). (31c) The ( p) 1 (ˆε t, ) 2w = ( p) 1 (A t, + B t, + C t, + D t, ) 2w. If we expad the right-had side of this last equatio, we will get a sum of terms of the form ( p) 1 A a t,b b t,c c t,d d t, where a, b, c, d 0 ad a + b + c + d = 2w. Next we apply Hölder s iequality to each of these terms. 20

21 The, because of (31), all these terms are o p (1) apart from the term ( p) 1 A2w t,. Hece, ( p) 1 (ˆε t, ) 2w = ( p) 1 (ε t, ) 2w + o p (1). We ca the establish (29) by applyig the weak law of large umbers. Fially, we expad the right-had side of (20) ad agai apply Hölder s iequality to the crossterms. The proof is the completed usig (29) ad (21). Lemma 2 Suppose Assumptios 1 (with r = 4 ad s = 1) ad 2 hold. The as : (a) there exists a radom variable 0 such that sup 0 j=0 j ˆψ j, < i probability; (b) sup 0 j ˆψ j, ψ j = o p (1); (c) Var [ε t,] σ 2 ε = o p (1) (d) Var [ 1/2 ε t, ] σ2 = o p (1). Proof of Lemma 2 (a) As show i Bühlma (1995, Lemma 2.2) it is sufficiet to prove that From the triagular iequality, we have j=0 j ˆφ j, φ j = o p (1). (32) j=0 j ˆφ j, φ j j ˆφ j, φ j, + j φ j, φ j. (33) j=0 Bühlma (1995, Proof of Lemma 3.1) shows that j=0 j φ j, φ j = o(1) a.s. ad furthermore we have that j ˆφ j, φ j, p 2 max ˆφ j, φ j, = o p (1). (34) 1 j p j=0 Hece, j=0 j ˆφ j, φ j = o p (1) ad the proof of part (a) is completed. Proof of Lemma 2 (b) See Bühlma (1995, Proof of Theorem 3.2) for the proof. j=0 Proof of Lemma 2 (c) ad (d) Usig Lemma 1 ad parts (a) ad (b) these results follow as i Psaradakis (2001, Proof of Lemma 2). Lemma 3 Let S (r) = 1/2 r u t, ad suppose Assumptios 1 (with r = 4 ad s = 1) ad 2 hold. The S (r) σw(r). (35) Proof of Lemma 3 See Psaradakis (2001, Proof of Lemma 3). Lemma 4 Let ξt, = t i=1 u t, ad let Assumptios 1 (with r = 4 ad s = 1) ad 2 hold. The 21

22 a 3/2 ξ t 1, σ 1 0 W(r)dr, b 2 ξ 2 t 1, σ2 1 0 W(r)2 dr, c 5/2 tξ t 1, σ 1 0 rw(r)dr, d 1 ξ t 1, u t, σ2 1 0 W(r)dW(r) (σ2 σ 2 u ), e 3/2 tu t, σ 1 0 rdw(r). Proof of Lemma 4 See Psaradakis (2001, Proof of Lemma A.1). Proof of Theorem 1 See Psaradakis (2001, Proof of Theorem 1). A.2 Proof of Theorem 2 Agai we first eed several lemmas. Lemma 5 Let Assumptio 1 (with r 4 ad s 1) hold ad let p() = o((/ l) 1/2 ). The it follows that (a) max 1 j p ˆφ j, φ j = O p ((l /) 1/2 ) + o(p s ) (b) ˆσ 2 = σ 2 + O p ((l/) 1/2 ) + o(p s ) (c) p ˆφ j, = φ j + O p (p(l/) 1/2 ) + o(p s ). Proof of Lemma 5 part (a) See Chag ad Park (2002, Lemma 3.5). Proof of Lemma 5 part (b) See Bühlma (1995, Proof of Theorem 3.2) Proof of Lemma 5 part (c) See Chag ad Park (2002, Lemma 3.5). Lemma 6 Let Assumptio 1 (with r 4 ad s 1) hold ad let p() = o((/ l) 1/2 ). The 1 r/2 E ε t, r p 0 ad W (i) = 1 [i] ˆσ k=1 ε k, d W(i) as. Proof of Lemma 6 As show i Park (2002, Theorem 2.2) we oly eed to show 1 r/2 E ε t, r = 1 r/2 ( 1 ˆε t, 1 Our proof will follow the lies of Park (2002, Proof of Lemma 3.2). We have that 1 ˆε t, 1 r) p ˆε t, 0. (36) r ˆε t, c(a + B + C + D ) (37) 22

23 where ad A = 1 C = 1 ε t r B = 1 ε t, ε t r ˆε t, ε t, r 1 r D = ˆε t, ε t, = y t ˆε t, is defied i (15). Furthermore, let φ j, be defied such that i y t = e t, is ucorrelated with y t 1,..., y t p. φ j y t j. (38) φ j, y t j + e t,, (39) Hece we have to show that 1 r/2 A, 1 r/2 B, 1 r/2 C ad 1 r/2 D p 0. The results for A ad B are show i Park (2002, Proof of Lemma 3.2). Next we tur to C. We write ˆε t, = y t ˆρ y t 1 ˆφ j, y t j = (y t ˆρ y t 1 y t ) + ( y t ˆφ j, y t j ) = (ˆρ 1)y t 1 + (ε t, (ˆφ j, φ j, ) y t j (φ j, φ j ) y t j ) (40) It the follows that ˆε t, ε t, r c (ˆρ 1)y t 1 r + (ˆφ j, φ j, ) y t j where c = 3 r 1. We defie r r + (φ j, φ j ) y t j. (41) C 0 = 1 (ˆρ 1)y t 1 r C 1 = 1 r (ˆφ j, φ j, ) y t j C 2 = 1 r (φ j, φ j ) y t j (42a) (42b) (42c) so that it eeds to be show that 1 r/2 C i a.s. 0 for i = 0, 1, 2. The result for C 2 follows from Park (2002, Proof of Lemma 3.2). 23

24 Agai followig Park (2002, Proof of Lemma 3.2), C 1 is majorised by ( ) 1 max ˆφ j, φ j, r 1 j p ( max ˆφ j, φ j, r 1 j p y t j r ( 1 ) p y t r + t=0 ( c max ( φ j, φ j, r + ˆφ j, φ j, r ) 1 j p 1 p y t r ) t= 1 ( 1 ) p y t r + t=0 1 p t= 1 y t r ) = [ O ((l /) r ) + O p ( r )] (p/)o() = O (p(l/) r ) = o p ((l/) r 1/2). (43) As r 4, C 1 p 0. Next we cosider C 0. Rewrite the expressio i (42a) for C 0 as 1 (ˆρ 1)y t 1 r = ˆρ 1 r 1 This proves that 1 r/2 C p 0. For D we eed to prove that 1 ˆε t, = 1 y t 1 r = o p (1). (44) ε t, + o p (1) = 1 which, by (40) ad the result that ε t, = ε t + j=p+1 φ j y t j, holds if j=p+1 j=p+1 j=p+1 ε t + o p (1), (45) φ j y t j p 0, (46) (φ j, φ j ) y t j p 0, (47) (ˆφ j, φ j, ) y t j p 0, (48) 1 p (1 ˆρ )y t 1 0, (49) where (46) ad (47) follow from Park (2002, Proof of Lemma 3.2). For (48) we defie N = (ˆφ j, φ j, ) y t j (50) ad Q = y t j. (51) 24

25 The N is domiated by Q max 1 j p ˆφ j, φ j,. (52) Park (2002, Proof of Lemma 3.2) shows that Q = o ( p 1/2 (l) 1/r (l l) (1+δ)/r) a.s. for ay δ > 0. Furthermore, max ˆφ j, φ j, max ( φ j, φ j, + ˆφ j, φ ( j, ) = O (l /) 1/2) ( a.s. + O p 1 ), (53) 1 j p 1 j p from which we ca coclude that N = o p (), which proves the result. Fially, from (25) it is easy to see (49) holds as well. This completes the proof of D ad hece of Lemma 6. Lemma 7 Let Assumptio 1 (with r 4 ad s 1) hold ad let p() = O((/ l) 1/3 ). The V (i) = 1 [i] k=1 u k, d σ j=0 ψ j W(i) as. Proof of Lemma 7 Give Lemma 2, see Park (2002, Proof of Theorem 3.3). Lemma 8 Let ω 2 = (1/) y t ad ω 2 = (1/) y t. Furthermore assume that Assumptio 1 holds with r 4 ad s 1. ad p() = o((/ l) 1/2 ). The we have for ay δ > 0, P[ ω 2 ω 2 δ] p 0. Proof of Lemma 8 See Park (2002, Proof of Lemma 4.1). Proof of Theorem 2 Give Lemmas 5 to 8, all the relevat lemmas foud i Chag ad Park (2003) are valid for the test with residuals. The proof the cocludes by Chag ad Park (2003, Proof of Theorem 2). 25

26 Table 1: Mai features of the tests. Test a Bootstrap Method Based o Test Statistic τ S,d Sieve Differeces DF τ t S,d Sieve Differeces DF t τ S,r Sieve Residuals DF τ t S,r Sieve Residuals DF t τs,d a Sieve Differeces ADF τ t a S,d Sieve Differeces ADF t τs,r a Sieve Residuals ADF τ t a S,r Sieve Residuals ADF t τ B,r Block Residuals DF τ τ B,d Block Differeces DF τ τb,r a Block Residuals ADF τ τb,d a Block Differeces ADF τ τ St,d Statioary Differeces DF τ t St,d Statioary Differeces DF t τ St,r Statioary Residuals DF τ t St,r Statioary Residuals DF t a We use τ for a coefficiet test ad t for a t-test. The first subscript idicates the bootstrap method: S stads for sieve bootstrap, B for block bootstrap, ad St for statioary bootstrap; the secod subscript idicates whether a test is based o differeces (d) or residuals (r). A superscript a states that the test is a augmeted DF test. 26

27 Pael A: Sieve tests τ S,d t S,d τ S,r t S,r τs,d a t a S,d τs,r a t a S,r Explaatory variables up to order O(1) L(P a ) (0.03) (0.03) (0.02) (0.02) (0.02) (0.01) (0.02) (0.01) Adj. R Explaatory variables up to order O( 1/2 ) L(P a ) (0.03) (0.03) (0.02) (0.02) (0.02) (0.01) (0.02) (0.02) Adj. R Explaatory variables up to order O( 1 ) L(P a ) (0.03) (0.04) (0.02) (0.02) (0.03) (0.01) (0.01) (0.01) Adj. R Pael B: Block-type tests τ B,r τ B,d τb,r a τb,d a τ St,d t St,d τ St,r t St,r Explaatory variables up to order O(1) L(P a ) (0.08) (0.05) (0.04) (0.05) (0.05) (0.05) (0.09) (0.09) Adj. R Explaatory variables up to order O( 1/2 ) L(P a ) (0.12) (0.06) (0.06) (0.12) (0.06) (0.06) (0.14) (0.13) Adj. R Explaatory variables up to order O( 1 ) L(P a ) (0.03) (0.05) (0.02) (0.05) (0.06) (0.05) (0.03) (0.03) Adj. R Table 2: Respose surfaces of size 27

28 τ S,d t S,d τ S,r t S,r τs,d a t a S,d τs,r a t a S,r Explaatory variables up to order O(1) L(P a ) (0.11) (0.11) (0.05) (0.05) (0.05) (0.04) (0.05) (0.04) Costat (0.36) (0.36) (ρ 1) (5.48) (5.39) (6.17) (6.14) (6.62) (5.48) (6.33) (5.14) (ρ 1) (27.79) (27.35) (31.37) (31.37) (31.61) (26.28) (30.68) (24.47) (ρ 1) 3 Adj. R Explaatory variables up to order O( 1/2 ) L(P a ) (0.04) (0.04) (0.02) (0.02) (0.17) (0.15) (0.15) (0.14) Costat (0.21) (0.21) (0.59) (0.52) (0.55) (0.52) (ρ 1) (6.46) (6.81) (5.80) (6.24) (9.02) (7.13) (8.22) (10.26) (ρ 1) (33.69) (35.75) (32.58) (35.33) (43.06) (35.43) (39.92) (106.11) (ρ 1) (327.03) Adj. R Explaatory variables up to order O( 1 ) L(P a ) (0.39) (0.40) (0.13) (0.02) (0.61) (0.51) (0.55) (0.50) Costat (1.37) (1.48) (0.51) (2.09) (1.84) (1.93) (1.78) (ρ 1) (8.11) (18.61) (14.28) (13.88) (9.43) (31.97) (8.73) (30.72) (ρ 1) (52.85) (133.58) (87.88) (92.06) (38.60) (239.34) (33.83) (231.53) (ρ 1) (165.86) (274.24) (493.75) (481.09) Adj. R Table 3: Respose surfaces of power - part I 28